As pointed out in the introduction transformations based on classical
Keplerian elements can only achieve a limited precision.
But for many applications it is useful to have approximate
positions available. For this reason we describe in the
following the calculation of position and velocity of
objects in Keplerian orbits. There are many textbooks on this
subject - we recommend Murray and Dermott (2000) but e.g. Bate et al. (1971),Danby (1988)
or Heafner (1999) are also very useful.
There are also some good web sites devoted to the
subject^{20}.

The gravitational motion of two bodies of mass and and position
vectors and can be described in terms of the three
invariants:
*gravitational parameter*
, *specific
mechanical energy*
, and
*specific angular momentum*
, where
,
and
.
is the constant of gravitation whose IAU1976 value is determined by [A. K6]:

(24) |

The elements of the conical orbit (shown in Fig.2) are then given as
*semi-major axis*
and *semi-minor axis*
,
or alternatively as *semi-latus rectum*
and
*eccentricity*
.
Let the origin be at the focus , the vector then describes the motion
of the body .
The *true anomaly* is the angle between and the direction to the closest point
of the orbit (*periapsis*)
and can be determined from

(25) |

*Mean elements* of a body in an elliptical orbit () are defined by the motion of a point
on a concentric circle with constant angular velocity
and radius , such that
the orbital period
is the same for and .
The *mean anomaly*
is defined as the angle between
periapsis and .
Unfortunately there is no simple relation between and the true anomaly .
To construct a relation one introduces another auxiliary concentric circle with radius
and defines as the point on that circle which has the same perifocal
x-coordinate as .
The *eccentric anomaly* is the angular distance between and the periapsis measured
from the centre and is related to the mean and true anomalies by the set of equations:

(27) |

Thus, if the orbital position is given as an expansion in of the
*mean longitude*
, the *true longitude*
can
be found by an integration of the transcendental Kepler equation.
In most cases a Newton-Raphson integration converges quickly (see Danby (1988) or Herrick (1971) for methods).
For hyperbolic orbits () one can as well define a mean anomaly
but this
quantity has no direct angular interpretation. The *hyperbolic eccentric anomaly*
is related to and the true anomaly by

The orientation of an orbit with respect to a reference plane (e.g. ecliptic) with origin at the
orbital focus is defined by the *inclination* of the orbital plane,
the *longitude of the ascending node* , and the *argument of
periapsis* which is the angle between ascending node and periapsis (see Fig.3).
The position of the body on the orbit can then be defined by
its *time of periapsis passage* , its true anomaly at epoch ,
or its true longitude
at epoch .
The *perifocal coordinate system* has its X-axis from the focus to the periapsis,
and its Z-axis right-handed perpendicular to the orbital plane in the sense of orbital motion.
In this system the position and velocity vector are given by

(29) |

In the hyperbolic case replace by and by . The transformation from the reference system to the perifocal system is given by the Eulerian rotation as defined in the Appendix. The ecliptic position of a planet is then given by .